Respuesta :
Answer:
The error bound is 3.125%.
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence interval [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
For this problem, we have that:
A sample of 506 California adults.. This means that [tex]n = 506[/tex].
76% of California adults (385 out of 506 surveyed) feel that education is one of the top issues facing California. This means that [tex]\pi = 0.76[/tex]
We wish to construct a 90% confidence interval
So [tex]\alpha = 0.10[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.10}{2} = 0.95[/tex], so [tex]z = 1.645[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.76 - 1.645\sqrt{\frac{0.76*0.24}{506}} = 0.7288[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.76 + 1.645\sqrt{\frac{0.76*0.24}{506}} = 0.7913[/tex]
The error bound of the confidence interval is the division by 2 of the subtraction of the upper limit by the lower limit. So:
[tex]EBM = \frac{0.7913 - 0.7288}{2} = 0.03125[/tex]
The error bound is 3.125%.
